"World Record ! 6.16 at my first attempt ! That's incredible, I'm still in the air," Lavillenie wrote on his official Twitter feed after the French Athletics Federation had flagged up the new record.
The previous mark of 6.15 was set by Sergei Bubka at the same event in Ukraine in 1993.
"It's completely unbelievable," Lavillenie, who took his head in his hands when he realised he had done it, later told French news channel BFM TV.
"I will need time to get back down on Earth. It was a mythical record. I knew I had the potential to do try it. But beat it so early, that's something else."
"I did not know what was happening to me. The sound was crazy. These are huge emotions. I am in a new dimension."
Six-time outdoor world champion and 1988 Olympic gold medallist Bubka still holds the outdoor world record of 6.14.
Bubka, who on Saturday morning had hoped Lavillenie would beat his record in his hometown of Donetsk if it had to happen one day, stood up and clapped his hands along with the cheering crowd after his mark was erased off the books.
"That's fantastic. It's history and I'm very happy that Renaud does it here, in Donetsk, my town," said Bubka, who came on the track to congratulate the Frenchman.
"I'm happy because my job is to help athletes perform. Athletics are my life. I'm pleased for him and for athletics," added Bubka, who is the vice-president of the International Association of Athletics Federations.
The 27-year-old Lavillenie had made the world record his priority of the season and started using longer poles last year.
He has kept improving his best indoor in the past weeks and became the second highest man in history two weeks ago in Bygdoszcz, Poland, with a 6.08 metres effort.
On Saturday, he went on the brink of a disappointment after two failures at 6.01 metres, only to eventually succeed and beat the record on his first chance.
Lavillenie, who won the Olympic title at the London Games and his first indoor world title the same year, has an outdoor best of 6.02 metres.
He has so far failed to win an outdoor world title, collecting one silver and two bronze medals.
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